Comparison of different test day models used for genetic evaluation of Polish Black-and-White cattle

A single-trait random regression (RR) model was used for a data set of 151,413 test day milk yields of 22,068 cows calving for the first time from 1992 through 1994. The fixed regressions were considered within 32 subclasses of genetic group (defined according to the proportion of HF genes) by age group by season of calving. Two different submodels were used for fixed and random regressions: regression model (RM) and Wilmink model (WM) as well as HYS or HTD fixed effects were incorporated, giving eight different models for comparison. Differences in estimated breeding values (BVs) calculated on the basis of solutions from each of eight RR models indicated that the change of HYS to HTD effect had no significant influence on BVs. Residual variances were reduced by up to 23% when models with HTD effects were used, showing that they accounted for factors specific to each test day much better than did models with HYS. Correlations between BVs from test day models and the national 305d BVs were 0.71-0.79 for sires and were higher for models with HYS effect. Comparison of the correlations between BVs from different test day models indicated that the choice of covariates for random regressions was important. The use of the Wilmink function for those regressions is justified because it is accurate enought and computationally more efficient.


INTRODUCTION
Accurate estimation of breeding values is of great importance in cattle breeding programmes.For many years genetic evaluation of dairy sires and cows has been based on analysis of 305d lactation milk production.Since 1991 there has been growing interest in using test day (TD) yields instead of the traditional 305d lactation yields for that purpose.In the first TD model, fixed regressions within age and season of calving were included to account for the lactation curve of groups of cows (Ptak and Schaeffer, 1993).In the next TD model, random regressions were incorporated to model a separate lactation curve for every cow (Schaeffer and Dekkers, 1994).In recent years much work has been done to improve models for analysis of TD data (Jamrozik et al., 1997a(Jamrozik et al., ,b, 1998;;Van der Werf et al., 1998;Rekaya et al, 1999).
TD models account for genetic and environmental effects directly influencing milk yield at the day of recording.They use every piece of information available and there is no need for extension factors.No assumptions about the lactation course have to be made.Models for TD yields allow more accurate modeling of single yields and reduce the residual variance (Ptak and Schaeffer, 1993;Reents et al., 1995;Swalve, 1995;Ptak and Zamecki, 1998).
In TD models the environmental effects are often represented by herd-test day (HTD) effects rather than herd-year-season (HYS) effects, resulting in remarkable reduction of residual variance (Ptak and Schaeffer, 1993;Ptak and Zamecki, 1998).Ptak and Schaeffer (1993) compared TD models with HTD effects and with HYS effects finding the former more suitable than the latter for the Canadian dairy cattle population.In countries with small herd sizes, like Poland or Germany, the number of HTD subclasses is usually many times greater than that of HYS, adversely affecting the degree of connectedness and causing problems with estimability.Griinhagen and Swalve (1997) concluded that there was only a slight impact of the small size of HTD subclasses on the degree of connectedness and estimated breeding values.In order to increase the accuracy of genetic evaluation of dairy cattle, different submodels (as functions of covariates for fixed and random regressions) in TD models were tested (Jamrozik et al., 1997b;Guo and Swalve, 1997).Fewer covariates and simpler combinations of covariates are better for computations.
The objective of this study was to compare random regression (RR) models for milk yield with either HTD or HYS effects in a population with small herd sizes.The comparison also included the use of different submodels for fixed and random regressions.

MATERIAL AND METHODS
The data file consisted of 151,413 TD milk yields of 22,068 first lactation cows with an average of 7 tests per cow.Cows were calved from 1992 through 1994.There were 1,339 sires and 20,039 dams.The pedigree file contained 45,775 animals including seven phantom parent groups (three created for bulls and four for dams).The data were grouped in 4,661 HYS subclasses or 27,903 HTD subclasses, almost 27% of them with single records.Only data from cows that calved between 18 and 48 months of age and TD yields recorded between 5 and 305d in milk were used.
The following single-trait random regression model (Jamrozik et al., 1997a = (1, I exp^O.OSt,))' or RM-regression submodel (x jl0 , x jn , x. p , x jl3 , x jl4 )' = (1, t^OSXt./SOS) 2 , 1^305/1,), 1^(305/1,))'.G and R denote the (co)variance matrices for additive genetic and residual effects, A is the numerator relationship matrix between animals, and I is the identity matrix.R is the diagonal matrix, with elements equal to estimates of residual variances i.e.Rdiag (G\ These values (cT)were calculated for each TD record using regression equation with linear and quadratic terms.The regression equation was fitted to variances of errors (e. jk] ) in each day in milk through lactation to smooth these values.The residuals were evaluated as:

jm jlm ^j
The symbol ® denotes the Kronecker product function (Searle, 1982).
The data were analyzed using different combinations of fixed effect H (HYS or HTD) and different combinations of covariates for fixed and random regressions (RM-regression submodel (Ali and Schaeffer, 1987) or WM-Wilmink submodel (Wilmink, 1987)).As a result, eight models were applied to the TD data: (1) HYS effect and RM for fixed regressions and RM for random regressions, (2) HYS effect and WM for fixed regressions and RM for random regressions, (3) HYS effect and RM for fixed regressions and WM for random regressions, (4) HYS effect and WM for fixed regressions and WM for random regressions, (5) HTD effect and RM for fixed regressions and RM for random regressions, (6) HTD effect and WM for fixed regressions and RM for random regressions, (7) HTD effect and RM for fixed regressions and WM for random regressions, (8) HTD effect and WM for fixed regressions and WM for random regressions.
The fixed regressions were considered within 32 subclasses of genetic group by age group by season of calving, with numbers of TD yields per subclass in Table 1.The four genetic groups were pure Polish Black-and-White (PBW), ith <50% HF genes, with <75% HF genes, and with up to 100% HF genes.Four age groups were also created: 18 to 23 mo, 24 to 29 mo, 30 to 35 mo, and 36 to 48 mo.Cows calved in two seasons: summer (April-September) and winter (October-March).The BV based on TD yields for the j-th animal on each DIM were calculated as DBV.(DIM)= a j0 + a., (DIM/305) + a j2 (DIM/305) 2 + a j3 ln(305/DIM) + a j4 ln 2 (305/ DIM) for models (4 (2), ( 5) and ( 6), or as DBV (DIM)= a j0 + a^DIM + a j2 exp (-0.05DIM) for models (3), ( 4), ( 7) and ( 8), where a is a vector of solutions for random regression coefficients.The BVs for a complete 305d lactation were obtained by summing DBV(DIM) for DIM=1,305.These values were correlated with the national evaluations (BV(R)) based on first 305d lactation yields (Zamecki et al., 1999).
Comparisons of models for daily milk yields were based upon the differences in estimated breeding values (BV), residual variances, and correlations between actual and predicted TD yields.Correlations of BVs between pairs of models were also considered.

RESULTS AND DISCUSSION
Mean differences in BV for 305d milk yield between all pairs of models with HTD effect are plotted in Figure 1 for sires with more than 50 daughters.The mean differences between pairs of models with HYS effect were very similar to those shown in Figure 1, so for clarity they were not included in the plot, but they indicated that the change of HYS to HTD had not significant effect on BVs.The differences were smallest when the same submodel was used for both fixed and random regressions (model ( 5) and ( 8)) or when WM was used for fixed and RM for random regressions (model ( 6)).When RM was applied for fixed and WM for random regressions (model ( 7)), the differences from other submodels were very large, suggesting that this combination of functions used for regressions gave differently shaped lactation curves than did models ( 5), ( 6) and ( 8).In these cases the largest differences were observed.The standard deviations of the differences in BVs between models made up from 20% up to almost 850% of the mean.The smaller were mean of differences, the higher were values of the standard deviations of these differences.Jamrozik et al. (1997b) obtained similar results and concluded that changing the function used for the random regressions had a greater effect on the standard deviations of differences in BVs than did changing the function for fixed regressions.• ( 5)-(k) B( 6)-(k) D( 7)-(k) •( 8)-(k) Figure 1.Mean differences in BV between pairs of TD models with fixed effect of HTD for sires with more than 50 daughters Residual variance estimates calculated for each model are given in Table 2. Models with HTD effects accounted for factors specific to each test day much better than did those with HYS.Residual variances were reduced by 19 to 23% when models with HTD were used.Similar results were reported by Ptak and Schaeffer (1993).The use of regression (RM) or Wilmink (WM) submodels had no impact on the goodness of fit of models with either HYS or HTD effects.
Residual variance estimates for milk yields at every DIM through lactation for different models were calculated.For simplicity, only a few models were chosen to present the differences in residual variance estimates (Figure 2).Models with the same function for random regressions and different functions for fixed regressions followed almost the same pattern, indicating that the precision of the model is determined by the submodel chosen for random regressions (Jamrozik et al., 1997b;Ptak and Zamecki, 1998).The differences in residual variance estimates among models were negligible in 4, 7 and 9 months of lactation, whereas at the end of lactation they were large, especially for models with HYS.These variances ranged from 0.72 to 2.65 for models with HTD and from 0.96 to 5.08 for models with HYS (Figure 2).The correlations between actual and predicted TD yields are presented in Table 2.The higher values, 0.96, were observed in models with HTD effects (models ( 5)-( 8)).The correlations between actual and predicted TD yields within successive months of lactation were also calculated, and they ranged between 0.88 and 0.97.The fit of each model was better at the beginning of lactation (0.94-0.97); then it decreased, with the lowest values of correlations after 240 days in milk (0.88-0.95).
The correlations between BVs from TD models and the national 305d BVs were calculated for 79 sires with more than 50 daughters, and for 19,996 recorded cows (Table 3).In all cases, the correlations between BVs for bulls were higher than those between BVs for cows.Higher correlations between national BVs and TD BVs occured in models with HYS effects because these models were more similar to the model for national evaluation.In both types of models the HYS effect was defined as a common effect for all test day yields of all cows that calved in the same herd-year-season.
The correlations between BVs from TD models (1) and ( 2) as well as from models (3) and (4) was 0.99, indicating that the covariates for fixed regressions (RM or WM) gave the same solutions, whereas the correlations between BVs from other TD models were lower, and the choice of covariates for random regressions was important.This result agrees with those obtained by Jamrozik et al. (1997b).From a computational point of view it is better to choose the Wilmink submodel (WM) because it has only three parameters to estimate.The use of simpler functions as submodels in TD models has already been proposed by other authors (Guo and Swalve;1997, Jamrozik et al., 1997b)).
Comparison of all eight TD models indicates that models with HTD effects are better, and that the use of the Wilmink submodel for random regressions is justified because it is as accurate as the regression submodel but computationally more efficient.

(p) = Ia p , and
e. jkl is the random residual effect with var(e) -R.The vector of covariates, x." is defined as WM-Wilmink submodel (x.

TABLE 1
Number of test day records by age, genetic group and season of calving * PBW -Polish Black-and-Whites

TABLE 2
Residual variance estimates and correlations between actual and predicted test day yields for TD models Figure 2. Residual variance estimates of daily milk yields from some TD models